Equilibrium earnings management and managerial compensation in a multiperiod agency setting

Abstract

To investigate how the possibility of earnings manipulation affects managerial compensation contracts, we study a two-period agency setting in which a firm’s manager can engage in window-dressing activities to manipulate reported accounting earnings. Earnings manipulation boosts the reported earnings in one period at the expense of the reported earnings in the other period. We find that the optimal pay-performance sensitivity may increase and expected managerial compensation may decrease as the manager’s cost of earnings management decreases. When the manager is privately informed about the payoff of an investment project to the firm, we identify plausible conditions under which prohibiting earnings management can result in a less efficient investment decision for the firm and more rents for the manager.

Appendix

1.1 Proof of Lemma 1

Let G(β ₁,β ₂) denote the principal’s objective function in (5). It can be easily checked that G ₁₁ < 0, G ₂₂ < 0, and $G_{11}\cdot G_{22} - G_{12}^2 > 0$. Therefore G(β ₁,β ₂) is globally concave, and the optimal bonus coefficients $\beta_{t}^{\ast}$ are given by the following first-order conditions:

$$ \lambda_{1}^{2}-\beta_{1}^{\ast}\cdot\Upphi_1 -\frac{\left( \beta_{1}^{\ast}-\gamma\cdot\beta _{2}^{\ast}\right) }{w} =0, $$

(16)

$$ \lambda_{2}^{2}-\beta_{2}^{\ast}\cdot\Upphi_2 +\frac{\left( \beta_{1}^*-\gamma\cdot\beta_{2}^{\ast }\right) }{w} =0. $$

(17)

The result then follows by the solution to the above two linear equations in β ^*₁ and β ^*₂ .

1.2 Proof of Proposition 1

Let us first consider the case when $\beta_1^0 = \gamma \cdot \beta_2^0$. We note from (7) that $\beta_1^* - \gamma \cdot \beta_2^* = 0$. Substituting this in (16) and (17) reveals that:

$$ \lambda_t^2 - \beta_t^* \cdot \Upphi_t = 0, $$

and hence $\beta_t^* = \frac{\lambda_t^2}{\Upphi_t} = \beta_t^0.$ It thus follows that when $\beta_1^0 = \gamma \cdot \beta_2^0,$ the spread between the discounted bonus rates ($\beta_1^* - \gamma \cdot \beta_2^*$) and the average bonus rate (i.e., $\frac{\beta_1^* + \beta_2^*}{2}$) are both independent of w.

For the case when $\beta_1^0 \neq \gamma \cdot \beta_2^0$, we have:

$$ \frac{\partial\left( \left\vert \beta_{1}^{\ast}-\gamma\cdot\beta_{2}^{\ast }\right\vert \right) }{\partial w} = \frac{\partial m}{\partial w}> 0. $$

From the expressions for the optimal bonus coefficients in Lemma 1, we get:

$$ \beta_{1}^{\ast}+\beta_{2}^{\ast}=\frac{\left( 1+\gamma\right) \cdot\left( \lambda_{1}^{2}+\lambda_{2}^{2}\right) +w\cdot\lambda_{2}^{2}\cdot\Upphi _{1}+w\cdot\lambda_{1}^{2}\cdot\Upphi_2}{\gamma\cdot\Upphi_1+\Upphi _{2}+w\cdot\Upphi_1\cdot\Upphi_2}. $$

Differentiating with respect to w and simplifying yield:

$$ \frac{\partial\left( \beta_{1}^{\ast}+\beta_{2}^{\ast}\right) }{\partial w}=\frac{\left( \gamma\lambda_{2}^{2}\Upphi_1-\Upphi_2\lambda_{1} ^{2}\right) \cdot\left( \Upphi_1-\Upphi_2\right) }{\left( \gamma \cdot\Upphi_1+\Upphi_2+w\cdot\Upphi_1\cdot\Upphi_2\right) ^{2}}. $$

It thus follows that:

$$ \begin{aligned} sgn\left[ \frac{\partial\left( \beta_{1}^{\ast}+\beta_{2}^{\ast}\right) }{\partial w}\right] & =sgn\left[ \left( \gamma\lambda_{2}^{2}\Upphi _{1}-\Upphi_2\lambda_{1}^{2}\right) \cdot\left( \Upphi_1-\Upphi _{2}\right) \right] \\ & =sgn\left[ \left( \Upphi_2-\Upphi_1\right) \left(\beta_1^0 - \gamma \cdot \beta_2^0 \right) \right]. \end{aligned} $$

The conclusions in parts (i) and (ii) of Proposition 1 then follow.

1.3 Proof of Proposition 2

Recall that $|b^{\ast}|=\frac{|\beta_{1}^{\ast}-\gamma \cdot\beta_{2}^{\ast}|}{w}.$ Substituting for $\beta_{1}^{\ast}- \gamma \cdot \beta _{2}^{\ast}$ from (7) and simplifying yield:

$$ |b^{\ast}|=\frac{|\lambda_{1}^{2}\cdot\Upphi_2-\gamma\cdot\lambda_{2} ^{2}\cdot\Upphi_1|}{\gamma\cdot\Upphi_1+\Upphi_2+w\cdot\Upphi_1 \cdot\Upphi_2}. $$

It therefore follows that the equilibrium amount of earnings management $|b^{\ast}|$ is decreasing in w.

The firm’s expected profit can be expressed as:

$$ G(\beta_{1}^{\ast}(w),\beta_{2}^{\ast}(w),w)=\sum_{t=1}^{2} \gamma^{t} \cdot \left(\lambda_t^2 \cdot \beta_t^* - \frac{1}{2} \cdot {\beta_t^*}^2 \cdot \Upphi_t \right) -\frac{\gamma}{2}\cdot w^{-1}\cdot(\beta _{1}^{\ast}-\beta_{2}^{\ast})^{2}. $$

Differentiating with respect to w and applying the Envelope Theorem reveal that

$$ \frac{dG}{dw}=\frac{\gamma}{2}\cdot{w}^{-2}\cdot(\beta_{1}^{\ast}-\beta _{2}^{\ast})^{2}>0. $$

To prove the last part, we note that the present value of the manager’s expected compensation is given by:

$$ E(\gamma\cdot s_{1}+\gamma^{2}\cdot s_{2})=\frac{\gamma}{2}\cdot\lambda _{1}^{2}\cdot{\beta_{1}^{\ast}}^{2}+\frac{\gamma^{2}}{2}\cdot\lambda_{2} ^{2}\cdot{\beta_{2}^{\ast}}^{2}+\frac{\gamma}{2}\cdot\rho\cdot{\beta_{1} ^{\ast}}^{2}\cdot\sigma_{1}^{2}+\frac{\gamma^{2}}{2}\cdot\rho\cdot{\beta _{2}^{\ast}}^{2}\cdot\sigma_{2}^{2}+\frac{\gamma}{2}\cdot w^{-1}\cdot\left( \beta_{1}^{\ast}-\beta_{2}^{\ast}\right) ^{2}. $$

Substituting the expressions for the optimal bonus coefficients yields:

$$ \begin{aligned} & E(\gamma\cdot s_{1}+\gamma^{2}\cdot s_{2}) \\ & =\frac{\gamma\cdot\Upphi_1}{2}\cdot\left( \frac{\lambda_{2}^{2} +\lambda_{1}^{2}\left( 1+w\Upphi_2\right) }{\gamma \Upphi_1+\Upphi_2 +w\Upphi_1\Upphi_2}\right) ^{2}+\frac{\gamma^{2}\cdot\Upphi_2}{2} \cdot\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}\left( 1+w\Upphi _{1}\right) }{\gamma \Upphi_1+\Upphi_2+w\Upphi_1\Upphi_2}\right) ^{2} +\frac{\gamma\cdot w}{2}\cdot\left( \frac{\lambda_{1}^{2}\Upphi_2 -\lambda_{2}^{2}\Upphi_1}{\gamma \Upphi_1+\Upphi_2+w\Upphi_1\Upphi_2 }\right) ^{2} \\ & =\frac{\gamma}{2}\frac{\gamma\lambda_{1}^{4}+\gamma\lambda_{2}^{4} +2\gamma\lambda_{1}^{2}\lambda_{2}^{2}+w\Upphi_2\lambda_{1}^{4} +w\gamma\Upphi_1\lambda_{2}^{4}}{\gamma\cdot\Upphi_1+\Upphi_2 +w\cdot\Upphi_1\cdot\Upphi_2}. \end{aligned} $$

Differentiating with respect to w gives:

$$ \begin{aligned} \frac{d E(\gamma\cdot s_{1}+\gamma^{2}\cdot s_{2})}{d w} & =-\frac{\gamma\Upphi_1\Upphi_2\left( \gamma\left( \lambda_{1} ^{2}+\lambda_{2}^{2}\right) ^{2}+\left( \gamma\Upphi_1\lambda_{2} ^{4}+\Upphi_2\lambda_{1}^{4}\right) w\right) }{2\cdot\left( \gamma \cdot\Upphi_1+\Upphi_2+w\cdot\Upphi_1\cdot\Upphi_2\right) ^{2}} +\frac{\gamma\cdot\left( \Upphi_2\lambda_{1}^{4}+\gamma\Upphi_1 \lambda_{2}^{4}\right) }{2\cdot\left( \gamma\cdot\Upphi_1+\Upphi _{2}+w\cdot\Upphi_1\cdot\Upphi_2\right) } \\ & =\frac{\gamma\cdot\left( \gamma\Upphi_1\lambda_{2}^{2}-\Upphi_2 \lambda_{1}^{2}\right) ^{2}}{2\cdot\left( \gamma\cdot\Upphi_1+\Upphi _{2}+w\cdot\Upphi_1\cdot\Upphi_2\right) ^{2}} \\ & >0. \end{aligned} $$

This proves that the expected compensation is increasing in w.

1.4 Proof of Lemma 2

The objective function in (10) can be maximized pointwise. The optimal bonus coefficients will solve the following first-order conditions:

$$ \begin{aligned} \lambda_{1}^{2}-x_{1}\cdot H(\theta)\cdot I(\theta)-\beta_{1}^*(\theta )\cdot\Upphi_1-w^{-1}[\beta_{1}^*(\theta)-\gamma\cdot\beta_{2}^*(\theta)] & =0, \\ \lambda_{2}^{2}-x_{2}\cdot H(\theta)\cdot I(\theta)-\beta_{2}^*(\theta )\cdot\Upphi_2+w^{-1}[\beta_{1}^*(\theta)-\gamma\cdot\beta_{2}^*(\theta)] & =0. \end{aligned} $$

The above equations simplify to the first-order conditions in (16) and (17) for the case when I(θ) = 0, and hence $\beta_{t}^*(\theta)=\beta_{t}^{\ast}$ in the non-investment region. When I(θ) = 1, the above equations solve to yield the following optimal bonus coefficients:

$$ \beta_{1}^* \left( \theta\right) =\frac{\gamma\left( \lambda_{2}^{2} -x_{2}\cdot H\left( \theta\right) \right) +(\lambda_{1}^{2}-x_{1}\cdot H\left( \theta\right) )\left( \gamma+w\Upphi_2\right) }{\gamma \cdot\Upphi_1+\Upphi_2+w\cdot\Upphi_1\cdot\Upphi_2}, $$

(18)

and

$$ \beta_{2}^* \left( \theta\right) =\frac{\lambda_{1}^{2}-x_{1}\cdot H\left( \theta\right) +\left( \lambda_{2}^{2}-x_{2}\cdot H\left( \theta\right) \right) \left( 1+w\Upphi_1\right) }{\gamma\cdot\Upphi_1+\Upphi _{2}+w\cdot\Upphi_1\cdot\Upphi_2}. $$

(19)

Since $H(\cdot)$ is decreasing, it can be easily verified that $\beta _{t}^{\ast}(\theta)$ is increasing in θ for each t.

With respect to the investment policy, the principal will accept the project if and only if

$$ NPV\left( \theta\right) -\sum_{t=1}^{2}\gamma^{t}\cdot\beta_{t}^*(\theta)\cdot x_{t}\cdot H(\theta)+G(\beta_{1}^*(\theta),\beta_{2}(\theta))\geq G(\beta _{1}^{\ast},\beta_{2}^{\ast}). $$

(20)

Applying the Envelope Theorem, it can be easily verified that the left hand side of (20) is increasing in θ.

Note that $\beta_{t}^*(\bar{\theta})=\beta_{t}^{\ast}$, since $H(\bar{\theta })=0$. Hence (20) holds as an inequality at $\theta=\bar {\theta}$. On the other hand, the inequality in (20) fails to hold at θ⁰. Consequently, there exists a unique $\theta^{\ast}\in(\theta^{0},\bar{\theta})$ such that the principal invests if and only if $\theta\geq\theta^{\ast}$.

To complete the proof, we need to show that the resulting incentive scheme is globally incentive compatible. It is well known that an incentive scheme is incentive compatible provided it is locally incentive compatible (i.e., the incentive compatibility condition in (9) holds), and $\frac{\partial } {\partial \theta}CE(\hat \theta, \theta)$ is (weakly) increasing in $\hat \theta$. For the incentive scheme identified above,

$$ \frac{\partial }{\partial \theta}CE(\hat \theta, \theta) = I(\hat \theta) \cdot \sum_{t=1}^{2} \gamma^t \cdot x_t \cdot \beta_t^*(\hat \theta), $$

which is increasing in $\hat \theta$ since $\beta_t^*(\cdot)$ is increasing, and the optimal $I (\cdot)$ is an upper-tail investment policy.

1.5 Proof of Proposition 3

Using the expressions for the optimal bonus coefficients in (18) and (19), it can be shown that:

$$ \frac{\partial\left( \beta_{1}^*\left( \theta\right) +\beta_{2}^*\left( \theta\right) \right) }{\partial w}=\frac{\left[ \Upphi_1-\Upphi _{2}\right] \cdot\left[ \gamma\cdot\left( \lambda_{2}^{2}-H\left( \theta\right) \cdot x_{2}\right) \cdot\Upphi_1-\left( \lambda_{1} ^{2}-H\left( \theta\right) \cdot x_{1}\right) \cdot\Upphi_2\right] }{\left( \Upphi_2+\gamma\Upphi_1+w\Upphi_1\Upphi_2\right) ^{2}}. $$

Hence

$$ sgn\left[ \frac{\partial\left( \beta_{1}^*\left( \theta\right) +\beta _{2}^*\left( \theta\right) \right) }{\partial w}\right] =sgn\left[ (\Upphi_1-\Upphi_2)\cdot\left( \gamma\cdot\beta_{2}^{0}(\theta)-\beta _{1}^{0}(\theta)\right) \right] . $$

The conclusion in Proposition 3 follows.

1.6 Proof of Proposition 4

Note that the optimal cutoff $\theta^{\ast}$ is given by:

$$ NPV\left( \theta^{\ast}\right) -[\gamma\cdot x_{1}\cdot\beta_{1}^*\left( \theta^{\ast}\right) +\gamma^{2}\cdot x_{2}\cdot\beta_{2}^*\left( \theta ^{\ast}\right) ]\cdot H\left( \theta^{\ast}\right) +G\left( \beta _{1}^*\left( \theta^{\ast}\right) ,\beta_{2}^*\left( \theta^{\ast}\right) \right) -G\left( \beta_{1}^{\ast},\beta_{2}^{\ast}\right) =0, $$

where

• (i) $G\left( \beta_{1},\beta_{2}\right) \equiv\sum_{t=1}^{2}\gamma^{t}\cdot\left[ \lambda_{t}^{2}\cdot\beta_{t}-\frac{1}{2}\cdot \Upphi_t\cdot\beta_{t}^{2}\right] -\frac{\gamma}{2}\cdot w^{-1}\cdot\left( \beta_{1}-\beta_{2}\right) ^{2},$

• (ii) $\beta_{t}^*\left( \theta\right)$ uniquely maximizes $G\left( \beta_{1},\beta_{2}\right) -[\gamma\cdot x_{1}\cdot\beta_{1}+\gamma^{2}\cdot x_{2}\cdot\beta_{2}]\cdot H\left( \theta\right) ,$ and

• (iii) β ^* _t uniquely maximizes $G\left( \beta_{1},\beta _{2}\right)$.

Differentiating the above expression for $\theta^{\ast}$ with respect to w and using the Envelope Theorem yield:

$$ \begin{aligned} & \left[ \gamma\cdot x_{1}+\gamma^{2}\cdot x_{2}-\left( \gamma\cdot x_{1}\cdot\beta_{1}^*\left( \theta^{\ast}\right) +\gamma^{2}\cdot x_{2} \cdot\beta_{2}^*\left( \theta^{\ast}\right) \right) \cdot H^{\prime}\left( \theta^{\ast}\right) \right] \frac{d\theta^{\ast}}{d w} \\ & =\frac{\partial}{\partial w}[G\left( \beta_{1}^{\ast},\beta_{2}^{\ast }\right) -G\left( \beta_{1}^*\left( \theta^{\ast}\right) ,\beta_{2}^*\left( \theta^{\ast}\right) \right) ]. \end{aligned} $$

Since

$$ \frac{\partial G\left( \beta_{1},\beta_{2}\right) }{\partial w}=\frac {\gamma}{2}\cdot w^{-2}\cdot\left( \beta_{1}-\gamma\cdot\beta_{2}\right) ^{2}, $$

substituting for $\beta_{t}^{\ast}$ and $\beta_{t}\left( \theta^{\ast}\right)$ and simplifying yield:

$$ \frac{d\theta^{\ast}}{d w}=\frac{\gamma\left( \Upphi_1 \Upphi_2\right) ^{2}}{2\left( \gamma\Upphi_1+\Upphi_2+w\Upphi _{1}\Upphi_2\right) ^{2}}\left[ \frac{x_{1}}{\Upphi_1}-\frac{\gamma x_{2}}{\Upphi_2}\right] \frac{\left( 2\left( {\beta}_{1}^{0}-\gamma {\beta}_{2}^{0}\right) -H\left( \theta\right) \cdot\left( \frac{x_{1} }{\Upphi_1}-\frac{\gamma x_{2}}{\Upphi_2}\right) \right) H\left( \theta\right) }{\left[ \gamma x_{1}+\gamma^{2}x_{2}-\left( \gamma x_{1}\beta_{1}\left( \theta^{\ast}\right) +\gamma^{2}x_{2}\beta_{2}\left( \theta^{\ast}\right) \right) H^{\prime}\left( \theta^{\ast}\right) \right] }. $$

Since $H^{\prime}(\cdot)<0$, we get:

$$ sgn\left[ \frac{d \theta^{\ast}}{d w}\right] =sgn\left[ \left( \frac{x_{1}}{\Upphi_1}-\frac{\gamma\cdot x_{2}}{\Upphi_2}\right) \cdot\left\{ 2\left( {\beta}_{1}^{0}-\gamma\cdot{\beta}_{2}^{0}\right) -H\left( \theta^{\ast}\right) \left( \frac{x_{1}}{\Upphi_1}-\frac{\gamma \cdot x_{2}}{\Upphi_2}\right) \right\} \right] . $$

(21)

Since x ₂ = 1 − x ₁, we note that:

$$ sgn \left(\frac{x_1} {\Upphi_1} - \frac{\gamma x_2}{\Upphi_2}\right) = sgn(x_1 - \delta). $$

It thus follows that $\frac{d \theta^*}{d w} = 0$ when x ₁ = δ. When x ₁ < δ, the term inside the curly brackets on the right hand side of (21) is positive because $\beta_1^0 > \gamma \cdot \beta_2^0$. Hence $\frac{d \theta^*}{d w} < 0$ when x ₁ < δ.

Consider now the case when x ₁ > δ. Let ψ(x ₁) denote the term inside the curly brackets on the right hand side of (21). Note that when x ₁ > δ, $sgn \left [\frac{d \theta^*}{d w} \right] = sgn [\psi(x_1)].$ If $[\beta_1^0 - \gamma \cdot \beta_2^0] > \frac{H(\theta^*)} {2 \cdot \Upphi_1},$ we note that ψ(x ₁) is positive for x ₁ = 1 (which implies x ₂ = 0). Since $\psi(\cdot)$ is decreasing, this implies that ψ(x ₁) > 0 for all values of $x_1 \in [0,1].$ It therefore follows that $\frac{d \theta^*}{d w} > 0$ for all $x_1 \in (\delta,1].$

Suppose now $0 < [\beta_1^0 - \gamma \cdot \beta_2^0] < \frac{H(\theta^*)} {2 \cdot \Upphi_1}$. In this case, we note that ψ(1) < 0 and ψ(δ) > 0. Since $\psi(\cdot)$ is monotonically decreasing, this implies that there exists a unique cutoff $\bar x$ such that $\frac{\partial \theta^*}{\partial w} > 0$ for all $x_1 \in (\delta,\bar x)$ and $\frac{d \theta^*}{dw} < 0$ for all $x_1 > \bar x$.

This completes the proof of Proposition 4.

1.7 Proof of Proposition 5

Let

$$ R\equiv{\int\limits_{\theta^{\ast}}^{\bar{\theta}}}\left[ \gamma\beta _{1}^*\left( \theta\right) x_{1}+\gamma^{2}\beta_{2}^*\left( \theta\right) x_{2}\right] \left[ 1-F\left( \theta\right) \right]\,d\theta $$

denote the manager’s expected information rents. Differentiating with respect to w yields:

$$ \frac{\partial R}{\partial w}={\int\limits_{\theta^{\ast}}^{\bar{\theta}} }\frac{\partial\left[ \gamma\beta_{1}^*\left( \theta\right) x_{1}+\gamma ^{2}\beta_{2}^*\left( \theta\right) x_{2}\right] }{\partial w}\left( 1-F\left( \theta\right) \right)\,d\theta-\frac{\partial\theta^{\ast} }{\partial w}\cdot\left[ \gamma\beta_{1}^*\left( \theta^{\ast}\right) \cdot x_{1}+\gamma^{2}\beta_{2}^*\left( \theta^{\ast}\right) \cdot x_{2}\right] \left( 1-F\left( \theta^{\ast}\right) \right) . $$

Using the expressions (18) and (19) for the optimal bonus coefficients, it can be easily verified that:

$$ \frac{\partial\left[ \gamma\beta_{1}^*\left( \theta\right) \cdot x_{1} +\gamma^{2}\beta_{2}^*\left( \theta\right) \cdot x_{2}\right] }{\partial w}=\Upomega\cdot\gamma\cdot\left[ \frac{x_{1}}{\Upphi_1}-\frac{\gamma x_{2} }{\Upphi_2}\right] \left[ \left( {\beta}_{1}^{0}-\gamma{\beta}_{2} ^{0}\right) -H\left( \theta\right) \cdot\left( \frac{x_{1}}{\Upphi_1 }-\frac{\gamma x_{2}}{\Upphi_2}\right) \right] , $$

where, for brevity, we define $\Upomega=\frac{\left( \Upphi_1\Upphi _{2}\right) ^{2}}{\left( \gamma \Upphi_1+ \Upphi_2+w\cdot\Upphi_1\cdot\Upphi_2\right) ^{2}}.$

It thus follows that:

$$ \frac{\partial\theta^{\ast}}{\partial w}=\frac{\gamma\Upomega}{2}\left[ \frac{x_{1}}{\Upphi_1}-\frac{\gamma x_{2}}{\Upphi_2}\right] \frac{\left( 2\left( {\beta}_{1}^{0}-\gamma{\beta}_{2}^{0}\right) -H\left( \theta^{\ast }\right) \cdot\left( \frac{x_{1}}{\Upphi_1}-\frac{\gamma\cdot x_{2} }{\Upphi_2}\right) \right) H\left( \theta^{\ast}\right) }{\left[ \gamma x_{1}+\gamma^{2}x_{2}-\left( \gamma x_{1}\beta_{1}^*\left( \theta^{\ast }\right) +\gamma^{2}x_{2}\beta_{2}^*\left( \theta^{\ast}\right) \right) H^{\prime}\left( \theta^{\ast}\right) \right] }. $$

When x ₁ = δ, we have $\frac{x_{1}}{\Upphi_1}=\frac{\gamma\cdot x_{2}}{\Upphi_2}\equiv \frac{\gamma}{\gamma\Upphi_1+\Upphi_2}.$ Therefore,

$$ \begin{aligned} \frac{\partial R}{\partial w} & =0{\hbox {, and}} \\ \sum_{i=1}^{2}\gamma^{i}\cdot x_{i}\cdot\beta_{i}^*\left( \theta^{\ast}\right) & =\gamma^{2}\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\gamma\Upphi _{1}+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) }{\gamma\Upphi _{1}+\Upphi_2}\right). \end{aligned} $$

Now we study how $\frac{\partial R}{\partial w}$ changes with x ₁ at x ₁ = δ (i.e., at $\frac{x_{1}}{\Upphi_1}=\frac{\gamma\cdot x_{2}}{\Upphi_2}\equiv\frac{\gamma}{\gamma\Upphi_1+\Upphi_2}).$ Recall that we have assumed β ⁰₁  > γβ ⁰₂ without loss of generality. Since $\frac{x_{1}}{\Upphi_1}-\frac{\gamma\cdot x_{2}}{\Upphi_2}$ increases as x ₁ increases,

$$ \begin{aligned} & sgn \left. \left( \frac{\partial} {\partial x_1} \left[ \frac{\partial R}{\partial w} \right] \right\vert _{x_{1}=\delta} \right) \\ & =sgn\left( {\int\limits_{\theta^{\ast}}^{\bar{\theta}}}\left[ 1-F\left( \theta\right) \right] d\theta-\frac{\sum_{i=1}^{2}\gamma^{i}\cdot x_{i}\cdot\beta_{i}^*\left( \theta^{\ast}\right) \cdot H\left( \theta^{\ast }\right) }{\gamma^{2}\cdot\frac{\Upphi_1+\Upphi_2}{\gamma\Upphi _{1}+\Upphi_2}-\sum_{i=1}^{2}\gamma^{i}\cdot x_{i}\cdot\beta_{i}^*\left( \theta^{\ast}\right) \cdot H^{\prime}\left( \theta^{\ast}\right) } \cdot\left[ 1-F\left( \theta^{\ast}\right) \right] \right) \\ & =sgn\left( {\int\limits_{\theta^{\ast}}^{\bar{\theta}}}\left[ 1-F\left( \theta\right) \right] d\theta-\frac{\left( \frac{\lambda_{1} ^{2}+\lambda_{2}^{2}}{\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta ^{\ast}\right) }{\gamma\Upphi_1+\Upphi_2}\right) \cdot H\left( \theta^{\ast}\right) }{\frac{\Upphi_1+\Upphi_2}{\gamma\Upphi_1 +\Upphi_2}-\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\gamma\Upphi _{1}+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) }{\gamma\Upphi _{1}+\Upphi_2}\right) \cdot H^{\prime}\left( \theta^{\ast}\right) } \cdot\left[ 1-F\left( \theta^{\ast}\right) \right] \right) . \end{aligned} $$

For uniform distribution, $H^{\prime}\left( \cdot\right) =-1$. Hence,

$$ \begin{aligned} & sgn\left( {\int\limits_{\theta^{\ast}}^{\bar{\theta}}}\left[ 1-F\left( \theta\right) \right] d\theta-\frac{\left( \frac{\lambda_{1}^{2} +\lambda_{2}^{2}}{\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta^{\ast }\right) }{\gamma\Upphi_1+\Upphi_2}\right) \cdot H\left( \theta^{\ast }\right) }{\frac{\Upphi_1+\Upphi_2}{\gamma\Upphi_1+\Upphi_2}-\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}}{\gamma\Upphi_1+\Upphi_2} -\frac{H\left( \theta^{\ast}\right) }{\gamma\Upphi_1+\Upphi_2}\right) \cdot H^{\prime}\left( \theta^{\ast}\right) }\cdot\left[ 1-F\left( \theta^{\ast}\right) \right] \right) \\ & =-sgn\left[ \frac{\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}} {\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) } {\gamma\Upphi_1+\Upphi_2}\right) }{\frac{\Upphi_1+\Upphi_2} {\gamma\Upphi_1+\Upphi_2}+\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2} }{\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) } {\gamma\Upphi_1+\Upphi_2}\right) }-\frac{{\int\nolimits_{\theta^{\ast} }^{\bar{\theta}}}\left( \bar{\theta}-\theta\right) d\theta}{\left( \bar{\theta}-\theta^{\ast}\right) ^{2}}\right] \\ & =-sgn\left[ \frac{\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2}} {\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) } {\gamma\Upphi_1+\Upphi_2}\right) }{\frac{\Upphi_1+\Upphi_2} {\gamma\Upphi_1+\Upphi_2}+\left( \frac{\lambda_{1}^{2}+\lambda_{2}^{2} }{\gamma\Upphi_1+\Upphi_2}-\frac{H\left( \theta^{\ast}\right) } {\gamma\Upphi_1+\Upphi_2}\right) }-\frac{1}{2}\right] >0. \end{aligned} $$

Hence $\frac{\partial}{\partial x_1}\left[\frac{\partial R} {\partial w}\right] > 0$ when evaluated at x ₁ = δ. Since $\frac{\partial R}{\partial w} = 0$ at x ₁ = δ, it follows that $\frac{\partial R}{\partial w}$ is positive (negative) for values of x ₁ greater (less) than δ in a neighborhood of x ₁ = δ. This completes the proof of Proposition 5.

1.8 Proof of Proposition 6

$$ \begin{aligned} b^{\ast}(\theta) & =w^{-1}\cdot\left[ \beta_{1}^*\left( \theta\right) -\gamma\cdot\beta_{2}^*\left( \theta\right) \right]\\ & =w^{-1}\cdot m\cdot\left[ (\beta_{1}^{0}-\gamma\cdot\beta_{2} ^{0})-H(\theta)\cdot\left( \frac{x_{1}}{\Upphi_1}-\frac{\gamma\cdot x_{2} }{\Upphi_2}\right) \right] , \end{aligned} $$

where m > 0 is as defined in connection with (7).

Since $H^{\prime}\left( \theta \right) <0$, we have:

$$ sgn\left( \frac{d b^{\ast}\left( \theta\right) }{d\theta }\right) =sgn\left( \frac{x_{1}}{\Upphi_1}-\frac{\gamma\cdot x_{2}} {\Upphi_2}\right) = sgn (x_1 -\delta). $$